Problem 2
A configuration of points in the plane is called Colombian if it consists of red points and blue points, and no three of the points of the configuration are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is good for a Colombian configuration if the following two conditions are satisfied: (1) no line passes through any point of the configuration; (2) no region contains points of both colours. Find the least value of such that for any Colombian configuration of points, there is a good arrangement of lines.
Step 1 of 5: Lower bound from an alternating convex polygon
In plain words
Every red-blue boundary gap must be crossed, and one line can cross the polygon boundary at most twice.
Detailed analysis
Place 2013 red and 2013 blue points alternately on a convex 4026-gon, and place the extra blue point in general position elsewhere. The 4026 boundary edges have opposite-coloured endpoints, so every good arrangement must cross each edge. One line meets the polygon boundary at most twice, hence .